First of all, I don't understand the meaning of "Assume an appropriate interval I of definition".

Second, I tried to differnetiate y = -(cos x) ln (sec x + tan x) twice in order to get y'' and replace it in the function but I couldn't. It's realy difficult and terribly long. I think I'm doing it in the wrong way. Please help me in that.

The interval part is put there for you not,as not to worry about singularity points (and points in which the argument of the natural logarithm is negative) and simply do what you already did,viz.differentiate.So do not worry.Simply compute the second order derivative,even if ugly...